Some remarks on Gordin-Lifšic's condition for martingale approximations

TL;DR

This work analyzes when the Gordin-Lifšic condition $\sum_{k \ge 0} \| \mathbb{E}(X_k \mid \mathcal{F}_0)\|_2^2 < \infty$ guarantees martingale approximations for strictly stationary sequences, and how these approximations yield central limit theorems and functional limit results. It establishes precise equivalences and sufficiency criteria for $L^2$-MA, $L^2_0$-MA, $L^2$-MMA, and $L^2_0$-MMA, linking them to convergence of Gordin’s martingale series $D_n = P_0(X_0+S_n)$ and to conditions such as Hannan's and Maxwell-Woodroofe. The paper provides detailed results for semi-linear processes, showing that $L^2$-MA/$L^2_0$-MA are equivalent to the $L^2$-convergence of $\sum_{i\ge0} A_i$, with $D=\sum_{i\ge0} A_i$, and extends these insights to Hölder observables, giving criteria for $L^2$-MMA and $L^2_0$-MMA and their quenched limit properties. It also clarifies the hierarchy among GL, MWstrong, and Hannan-type conditions, presents counterexamples illustrating optimality, and demonstrates how these results apply to $\\\alpha$-dependent sequences and to semi-linear processes with Hölder functions, thereby broadening the scope of martingale-approximation techniques in dependent sequences.

Abstract

In this note, we study a condition introduced by Gordin and Lif{\v s}ic in 1981 to establish the Central Limit Theorem for additive functionals of stationary Markov chains with normal transition operator. In the more general setting of strictly stationary sequences satisfying the Gordin-Lif{\v s}ic condition, we give sufficient (and sometimes also necessary) conditions for partial sums to be approximated in L2 by a martingale with stationary increments. Various types of L2 approximations are described, leading to different versions of the central limit theorem (annealed, quenched, functional form...). The optimality of the conditions is discussed, and an application to the class of semi-linear processes is presented.

Theorems & Definitions (16)

• Lemma 2.1
• Proposition 2.1
• Corollary 2.1
• Lemma 3.1
• Proposition 3.1
• Remark 3.1
• Proposition 4.1
• Remark 4.1
• Corollary 4.1
• Proposition 4.2